With this Lemma in "Basic Algebraic Topology" by Shastri, the author want prove the De Rham Theorem.
Can anyone help me to prove the underline claim?
Explanation in the proof of De Rham Theorem
1
$\begingroup$
algebraic-topology
proof-explanation
smooth-manifolds
-
0Note that $U_i = f^{-1}(i-1,i+1)\subseteq f^{-1}[i-1,i+1]$ and this last set is compact because $f$ is proper. – 2017-02-26
-
0Yes, this is clear...my problem is how to prove that $U_i$ is finite union of convex subsets. – 2017-02-26
-
1I see your concern now. It is not clear to me either. Try to see the proof of de Rham's theorem on Lee's Introduction to Smooth Manifolds (Th. 18.14). It is in the same spirit but it is a bit more explicit. – 2017-02-26